Neural Network Backstepping Controller Design for Fractional-Order Nonlinear Systems

In this work, a backstepping controller design for fractional-order strict feedback systems is investigated and the neural network control method is used. It is noted that in the standard backstepping design, the fractional derivative of the virtual quantity needs to be calculated repeatedly, which will lead to a sharp increase in the number of controller terms with the increase of the system dimension and finally make the control system difficult to bear. To handle the estimation error, certain robust terms in the controller at the last step are designed.)e stability of the controlled system is proven strictly. In addition, the proposed controller has a simple formwhich can be easily implemented. Finally, in order to verify our theoretical method, the control simulation based on a fractional-order chaotic system is implemented.


Introduction
Backstepping design is a recursive design method. It is a systematic design method for systems with uncertain parameters first proposed by Kanellakopoulos et al. in 1991 [1]. Its main ideas of this control method are to obtain a feedback controller by recursively constructing Lyapunov function of closed-loop system, select proper control law by deriving the Lyapunov function along the trajectory of closed-loop system, ensure the boundedness of closedloop system trajectory, and guarantee the convergence of the tracking error. e backstepping design method is suitable for both linear and nonlinear systems, so it has been widely used as soon as proposed. In addition, the backstepping method can be used together with many other control methods, such as sliding mode control [2,3], adaptive fuzzy control (AFC) [4,5], and intermittent control [6], and it is very effective in controlling the strict feedback systems. Backstepping control has some advantages, such as global stability, easy design, and implementation of the controller. However, it also has a big weakness named "explosion of terms," which is caused by repeatedly deriving the virtual controller. In fact, it is difficult to solve this problem completely. Meanwhile, it is well known that system uncertainties exist in most realworld systems. Hence, it is meaningful but difficult to investigate the backstepping control of uncertain nonlinear systems avoiding explosion of terms.
On the contrary, fractional calculus has almost the same history as the integer one. However, due to the complexity of the theory and the lack of corresponding physical background, it has not been developed as it should be. In 1983, Mandelbrot pointed out that there are a large number of fractional dimension phenomena in nature, and thus, fractional calculus has rapidly become a research hotspot. Studies show that many physical systems exhibit fractional-order dynamic behavior due to their special material and chemical properties, which are called fractional-order systems. Using fractional-order models to describe objects with fractional-order characteristics can better reveal the essential characteristics and behavior of objects [7]. Fractional calculus is the extension of integer calculus, and integral calculus is a special case of fractional calculus, so it is more universal to study fractional-order systems [8][9][10]. Compared with the integer-order model, the fractional-order model has clearer physical meaning and more concise expression in describing complex physical and mechanical problems; fractional-order control expands the degree of freedom of control and can obtain better control performance; fractional-order calculus has memory performance which ensures the influence of historical information on the present and future, and is conducive to improving the quality of control [11][12][13][14]. In the field of integer-order systems, based on Lyapunov stability theory, the control of nonlinear systems has been widely studied and a series of results have been obtained. However, due to the late start of fractional-order control and the complexity of the theory, the fractional-order stability theory and controller design method are far less developed than the control of integer-order systems.
In the last several decades, the nonlinear system control based on universal function approximators (neural networks and fuzzy system) has received much attention [4,[14][15][16][17][18]. It is shown that universal function approximators show an excellent performance in approximating system uncertainties.
us, backstepping control based on universal function approximators is an interesting research idea. In this aspect, many prior works have been given. In [4], fuzzy backstepping controller was designed for fractional-order systems, where the fractional virtual inputs are approximated by fuzzy systems. In [19], optimal backstepping control implemented for fractional-order systems was investigated. In [20], fuzzy backstepping control of strict-feedback fractional-order uncertain nonlinear systems was studied, and some interesting results were given. In [5], by using optimal control, type-2 fuzzy control, and sliding mode control, backstepping control of fractional systems is addressed, and it is shown that the backstepping control can be well mixed with many other control strategies. In [21], fuzzy backstepping control of fractional-order systems is addressed, and the saturation phenomenon in fractional systems was also studied. In [22], to avoid the chattering phenomenon in the traditional backstepping control, a novel control method was developed. Some recent works can be referred to [23][24][25][26][27][28] and the references therein.
e key idea of this paper is to try to solve the problem of "term explosion" in the backstepping control of fractional-order systems. Backstepping control technique is employed in the controller design. It should be emphasized that no prior knowledge about the system uncertainties and the unknown part of the fractional-order system is needed, and the term explosion problem is also solved by using neural networks to approximate the system uncertainties.

Problem Description.
In this paper, we will consider the fractional-order nonlinear systems listed as . . , n being unknown functions, and u(t) representing the input. Apparently, system (1) represents a large class of strict feedback fractional-order systems.
Our aim is to design a proper controller such that the system output y(t) tracks the desired trajectory ζ d (t) with certain accuracy. Since f i (ζ i ) is assumed to be unknown, it can be approximated by neural networks, which will be introduced later in this section. In addition, to meet our objective, we need the following assumptions.

Assumption 1.
e referenced function ζ d is smooth and is known in advance.

Assumption 2.
e signal ζ is always known.

Neural Network Description.
In this part, we will give a brief description of a multilayer network that will be used later. e structure of the used neural networks is depicted in Figure 1.
In real-world applications, a neural network is represented in the following form: , n, h, and m are the number of neurons in input layer, hidden layer, and output layer, respectively, y k denotes the output, and v ji is selected randomly on the interval [−1, 1]. According to [29][30][31], φ(·) is chosen as us, we have

Complexity
Generally, we can used the above neural network to estimate a nonlinear function as with ε(ζ(t)) being the optimal approximation error vector. W * is defined by To facilitate the controller design, we need the following assumption.

Fractional Calculus.
In this paper, the Caputo fractional-order derivative will be used, which is expressed by with 0 < α ≤ 1 being the order and Γ(·) representing Euler's function. In the following, we will use D α t to replace C 0 D α t for convenience.
e following results about the fractional calculus will be used.
Lemma 2 (see [21]). Suppose that z(t) ∈ R n is a smooth function. e following inequality holds:

Main Results
In this part, the controller design together with the stability analysis will be given based the backstepping design procedure. To meet the control aim, let with α i being a virtual input to be designed. e controller can be designed recursively because in the traditional backstepping design, virtual input α i+1 (t) depends on α i (t) and its derivative. us, the whole design process can be divided into n steps.
Step 1. According to the first equation in (11), we can obtain with representing an unknown function to be approximated. en, c 1 (ζ 1 (t)) can be approximated using neural network (5) as Denote W * � arg min It should be emphasized that W * 1 is a constant vector whose exact value is not needed in the controller design. Let where c(ζ 1 (t), W * 1 ) � W * 1 ψ(ζ 1 (t)). Like that, in the literature [32][33][34], it is reasonable to assume that ζ 1 (t) is bounded, i.e., |ε 1 (ζ 1 (t))| < ε 1 with ε 1 being an unknown constant. us, it is easy to obtain that Using (16), (17) can be written as Substituting (15) and (18) into (17), we have us, the virtual control α 1 is given by Complexity with k 1 > 0 being a constant.
Step 2. i, 2 ≤ i ≤ n − 1. We can design with c i (t) being the estimation of Step n. At this last step, we can construct the following controller: with u c being a supervisory input which is used to handle the approximation errors of neural networks. To meet the control aim, the robust control term is given by with k 1j > 0, j � 1, 2, . . . , n being proper design parameters satisfying certain conditions (see, the conditions in eorem 1).
In the controller design, neural networks' parameters are adjusted as with λ i > 0 being a design parameter. e following theorem is presented to show the controlled system's stability.

Theorem 1. Consider system (1) satisfying Assumptions 1-3.
If the virtual inputs are given by (20) and (21) and the final controller is given by (23) with a robust term (24), then the tracking error z 1 converges to origin asymptotically, and all signals remain bounded in the control process.
Let f(ζ 1 (t)) � 4.5ζ 1 e ζ 2 1 (t) , f(ζ 1 (t), ζ 2 (t)) � 3ζ 2 1 (t)+ ζ 3 2 (t), k 1 � k 2 � k 11 � k 12 � 0.95, and λ 1 � λ 5 � 25. Define the referenced signal ζ d (t) as where ζ c (t) is given by In the simulation, we should use 3 neural networks, whose nodes are 9, 81, and 729, respectively. e simulation results are shown in Figures 3-7 . Figure 3 mainly depicts the performance of signal ζ 1 (t) tracking the signal ζ d (t) which is defined by (36) and (37). From this figure, we can see that in the first 15 seconds, that is, when ζ d (t) is a constant, the tracking performance is good. When the time t is more than 15 seconds, the reference signal ζ d (t) has a sudden change, but after control, the output of the system can track ζ d (t) in a very short time. is also shows that the method has very good robustness. Figure 4 gives the virtual inputs α 1 (t) and α 1 (t). Figure 5 depicts the final controller u(t). From the above two pictures, we can see that all the inputs are smooth. Figure 6 shows the tracking errors z 1 (t), z 2 (t), and z 3 (t). Finally, Figure 7 depicts the norm of W 1 (t), W 2 (t), and W 3 (t). Generally speaking, our  Tracking results   6 Complexity method has good tracking performance, can ensure the tracking error convergence quickly, and has good robustness.

Conclusion
is paper concerns the control problem uncertain fractional-order nonlinear systems by using neural network backstepping technique. First, in the backstepping design, all virtual control inputs' fractional-order derivatives are approximated together with the system uncertainties by using 3-layer neural networks. By using this method, we need not calculate the fractional-order derivative of virtual input repeatedly and thus the problem of "explosion of terms" is overcome. To handle the approximation error, a robust term is needed. Finally, simulation studies illustrate the effectiveness of the proposed method. Noting that the nonlinear input is not considered in this work, how to solve this problem is one of our future research directions.

Data Availability
All datasets generated for this study are included in the manuscript.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.